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Theorem mulgfval 17714
Description: Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgval.b 𝐵 = (Base‘𝐺)
mulgval.p + = (+g𝐺)
mulgval.o 0 = (0g𝐺)
mulgval.i 𝐼 = (invg𝐺)
mulgval.t · = (.g𝐺)
Assertion
Ref Expression
mulgfval · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Distinct variable groups:   𝑥,𝑛, 0   𝑛,𝐺,𝑥   𝑛,𝐼,𝑥   𝐵,𝑛,𝑥
Allowed substitution hints:   + (𝑥,𝑛)   · (𝑥,𝑛)

Proof of Theorem mulgfval
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.t . 2 · = (.g𝐺)
2 eqidd 2749 . . . . 5 (𝑤 = 𝐺 → ℤ = ℤ)
3 fveq2 6340 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 mulgval.b . . . . . 6 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2800 . . . . 5 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6 fveq2 6340 . . . . . . 7 (𝑤 = 𝐺 → (0g𝑤) = (0g𝐺))
7 mulgval.o . . . . . . 7 0 = (0g𝐺)
86, 7syl6eqr 2800 . . . . . 6 (𝑤 = 𝐺 → (0g𝑤) = 0 )
9 seqex 12968 . . . . . . . 8 seq1((+g𝑤), (ℕ × {𝑥})) ∈ V
109a1i 11 . . . . . . 7 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) ∈ V)
11 id 22 . . . . . . . . . 10 (𝑠 = seq1((+g𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g𝑤), (ℕ × {𝑥})))
12 fveq2 6340 . . . . . . . . . . . 12 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
13 mulgval.p . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13syl6eqr 2800 . . . . . . . . . . 11 (𝑤 = 𝐺 → (+g𝑤) = + )
1514seqeq2d 12973 . . . . . . . . . 10 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
1611, 15sylan9eqr 2804 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
1716fveq1d 6342 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
18 simpl 474 . . . . . . . . . . 11 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
1918fveq2d 6344 . . . . . . . . . 10 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = (invg𝐺))
20 mulgval.i . . . . . . . . . 10 𝐼 = (invg𝐺)
2119, 20syl6eqr 2800 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = 𝐼)
2216fveq1d 6342 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
2321, 22fveq12d 6346 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → ((invg𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
2417, 23ifeq12d 4238 . . . . . . 7 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
2510, 24csbied 3689 . . . . . 6 (𝑤 = 𝐺seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
268, 25ifeq12d 4238 . . . . 5 (𝑤 = 𝐺 → if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛)))) = if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
272, 5, 26mpt2eq123dv 6870 . . . 4 (𝑤 = 𝐺 → (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
28 df-mulg 17713 . . . 4 .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))))
29 zex 11549 . . . . 5 ℤ ∈ V
30 fvex 6350 . . . . . 6 (Base‘𝐺) ∈ V
314, 30eqeltri 2823 . . . . 5 𝐵 ∈ V
3229, 31mpt2ex 7403 . . . 4 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V
3327, 28, 32fvmpt 6432 . . 3 (𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
34 fvprc 6334 . . . 4 𝐺 ∈ V → (.g𝐺) = ∅)
35 eqid 2748 . . . . . . 7 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
36 fvex 6350 . . . . . . . . 9 (0g𝐺) ∈ V
377, 36eqeltri 2823 . . . . . . . 8 0 ∈ V
38 fvex 6350 . . . . . . . . 9 (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
39 fvex 6350 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
4038, 39ifex 4288 . . . . . . . 8 if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
4137, 40ifex 4288 . . . . . . 7 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
4235, 41fnmpt2i 7395 . . . . . 6 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)
43 fvprc 6334 . . . . . . . . . 10 𝐺 ∈ V → (Base‘𝐺) = ∅)
444, 43syl5eq 2794 . . . . . . . . 9 𝐺 ∈ V → 𝐵 = ∅)
4544xpeq2d 5284 . . . . . . . 8 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
46 xp0 5698 . . . . . . . 8 (ℤ × ∅) = ∅
4745, 46syl6eq 2798 . . . . . . 7 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
4847fneq2d 6131 . . . . . 6 𝐺 ∈ V → ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅))
4942, 48mpbii 223 . . . . 5 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅)
50 fn0 6160 . . . . 5 ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅ ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
5149, 50sylib 208 . . . 4 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
5234, 51eqtr4d 2785 . . 3 𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
5333, 52pm2.61i 176 . 2 (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
541, 53eqtri 2770 1 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1620  wcel 2127  Vcvv 3328  csb 3662  c0 4046  ifcif 4218  {csn 4309   class class class wbr 4792   × cxp 5252   Fn wfn 6032  cfv 6037  cmpt2 6803  0cc0 10099  1c1 10100   < clt 10237  -cneg 10430  cn 11183  cz 11540  seqcseq 12966  Basecbs 16030  +gcplusg 16114  0gc0g 16273  invgcminusg 17595  .gcmg 17712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cnex 10155  ax-resscn 10156
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-neg 10432  df-z 11541  df-seq 12967  df-mulg 17713
This theorem is referenced by:  mulgval  17715  mulgfn  17716  mulgpropd  17756
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